A Finite Element Primer for Beginners : The Basics / / by Tarek I. Zohdi

• Autore: Zohdi Tarek I
• Pubblicazione: Cham : , : Springer International Publishing : , : Imprint : Springer, , 2018
• Edizione: 2nd ed. 2018.
• Descrizione fisica: 1 online resource (XIII, 135 p. 41 illus.)
• Disciplina: 620.00151535
• Soggetto topico: Mechanics; Mechanics, Applied; Computer mathematics; Computational complexity; Physics; Fluid mechanics; Mathematical models; Solid Mechanics; Computational Science and Engineering; Complexity; Numerical and Computational Physics, Simulation; Engineering Fluid Dynamics; Mathematical Modeling and Industrial Mathematics
• Nota di contenuto: Intro -- Preface -- Contents -- List of Figures -- 1 Weighted Residuals and Galerkin's Method for a Generic 1D Problem -- 1.1 Introduction: Weighted Residual Methods -- 1.2 Galerkin's Method -- 1.3 An Overall Framework -- 2 A Model Problem: 1D Elastostatics -- 2.1 Introduction: A Model Problem -- 2.2 Weak Formulations in One Dimension -- 2.3 An Example -- 2.4 Some Restrictions -- 2.5 Remarks on Nonlinear Problems -- 3 A Finite Element Implementation in One Dimension -- 3.1 Introduction -- 3.2 Weak Formulation -- 3.3 FEM Approximation -- 3.4 Construction of FEM Basis Functions -- 3.5 Integration and Gaussian Quadrature -- 3.5.1 An Example -- 3.6 Global/Local Transformations -- 3.7 Differential Properties of Shape Functions -- 3.8 Post-Processing -- 3.9 A Detailed Example -- 3.9.1 Weak Form -- 3.9.2 Formation of the Discrete System -- 3.9.3 Applying Boundary Conditions -- 3.9.4 Massive Data Storage Reduction -- 3.10 Quadratic Elements -- 4 Accuracy of the Finite Element Method in One Dimension -- 4.1 Introduction -- 4.2 The ``Best Approximation'' Theorem -- 4.3 The Principle of Minimum Potential Energy -- 4.4 Simple Estimates for Adequate FEM Meshes -- 4.5 Local Mesh Refinement -- 5 Iterative Solutions Schemes -- 5.1 Introduction: Minimum Principles and Krylov Methods -- 5.1.1 Numerical Linear Algebra -- 5.1.2 Krylov Searches and Minimum Principles -- 6 Weak Formulations in Three Dimensions -- 6.1 Introduction -- 6.2 Hilbertian Sobolev Spaces -- 6.3 The Principle of Minimum Potential Energy -- 6.4 Complementary Principles -- 7 A Finite Element Implementation in Three Dimensions -- 7.1 Introduction -- 7.2 FEM Approximation -- 7.3 Global/Local Transformations -- 7.4 Mesh Generation and Connectivity Functions -- 7.5 Warning: Restrictions on Elements -- 7.5.1 Good and Bad Elements: Examples -- 7.6 Three-Dimensional Shape Functions. ; 7.7 Differential Properties of Shape Functions -- 7.8 Differentiation in the Referential Coordinates -- 7.8.1 Implementation Issues -- 7.8.2 An Example of the Storage Scaling -- 7.9 Surface Jacobians and Nanson's Formula -- 7.10 Post-Processing -- 8 Accuracy of the Finite Element Method in Three Dimensions -- 8.1 Introduction -- 8.2 The ``Best Approximation'' Theorem -- 8.3 Simple Estimates for Adequate FEM Meshes Revisited for Three Dimensions -- 8.4 Local Error Estimation and Adaptive Mesh Refinement -- 8.4.1 A Posteriori Recovery Methods -- 8.4.2 A Posteriori Residual Methods -- 9 Time-Dependent Problems -- 9.1 Introduction -- 9.2 Generic Time Stepping -- 9.3 Application to the Continuum Formulation -- 10 Summary and Advanced Topics -- Appendix A Elementary Mathematical Concepts -- A.1 Vector Products -- A.2 Vector Calculus -- A.3 Interpretation of the Gradient of Functionals -- A.4 Matrix Manipulations -- A.4.1 Determinant -- A.4.2 Eigenvalues -- A.4.3 Coordinate Transformations -- Appendix B Basic Continuum Mechanics -- B.1 Deformations -- B.2 Equilibrium/Kinetics of Solid Continua -- B.2.1 Postulates on Volume and Surface Quantities -- B.2.2 Balance Law Formulations -- B.3 Referential Descriptions of Balance Laws and Nanson's Formula -- B.4 The First Law of Thermodynamics/An Energy Balance -- B.5 Linearly Elastic Constitutive Equations -- B.5.1 The Infinitesimal Strain Case -- B.5.2 Linear Elastic Constitutive Laws -- B.5.3 Material Component Interpretation -- B.6 Related Physical Concepts -- B.6.1 Heat Conduction -- B.6.2 Solid-State Diffusion-Reaction -- B.6.3 Conservation Law Families -- Appendix C Convergence of Recursive Iterative Schemes -- Appendix D Selected in-Class Exam Problems -- D.1 Sample Problem 1 -- D.2 Sample Problem 2 -- D.3 Sample Problem 3 -- D.4 Sample Problem 4 -- D.5 Sample Problem 5 -- D.6 Sample Problem 6. ; D.7 Sample Problem 7 -- D.8 Sample Problem 8 -- D.9 Sample Problem 9 -- D.10 Sample Problem 10 -- D.11 Sample Problem 11 -- D.12 Sample Problem 12 -- D.13 Sample Problem 13 -- D.14 Sample Problem 14 -- D.15 Sample Problem 15 -- D.16 Sample Problem 16 -- D.17 Sample Problem 17 -- Appendix E Selected Computer Projects -- E.1 Assignment Format -- E.2 Sample Project 1: The Basics of FEM -- E.3 Sample Project 2: Higher-Order Elements -- E.4 Sample Project 3: Potential and Efficient Solution Techniques -- E.5 Sample Project 4: Error Estimation and Adaptive Meshing Using the Exact Solution as a Test -- E.6 Sample Project 5: 3D Formulations for Elasticity -- E.7 Sample Project 6: Implementation of the Finite Element Method in 2D -- E.8 Sample Project 7: Time-Dependent Problems.
• Sommario/riassunto: The purpose of this primer is to provide the basics of the Finite Element Method, primarily illustrated through a classical model problem, linearized elasticity. The topics covered are: • Weighted residual methods and Galerkin approximations, • A model problem for one-dimensional linear elastostatics, • Weak formulations in one dimension, • Minimum principles in one dimension, • Error estimation in one dimension, • Construction of Finite Element basis functions in one dimension, • Gaussian Quadrature, • Iterative solvers and element by element data structures, • A model problem for three-dimensional linear elastostatics, • Weak formulations in three dimensions, • Basic rules for element construction in three-dimensions, • Assembly of the system and solution schemes, • An introduction to time-dependent problems and • An introduction to rapid computation based on domain decomposition and basic parallel processing. The approach is to introduce the basic concepts first in one-dimension, then move on to three-dimensions. A relatively informal style is adopted. This primer is intended to be a “starting point”, which can be later augmented by the large array of rigorous, detailed, books in the area of Finite Element analysis. In addition to overall improvements to the first edition, this second edition also adds several carefully selected in-class exam problems from exams given over the last 15 years at UC Berkeley, as well as a large number of take-home computer projects. These problems and projects are designed to be aligned to the theory provided in the main text of this primer. .
• Titolo autorizzato: A Finite Element Primer for Beginners
• ISBN: 3-319-70428-1
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910299585903321
• Lo trovi qui: Univ. Federico II